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Math 1271 Dis. 061 Quiz 8 November 15, 2005 Solutions No Work = No Credit. Write Legibly. Box your final result. 1. 10 points If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Since the base of the box is a square, we let x be the length of any side of the square. Thus, the area of the base is given by x2 . Let y be the height of the box (do not assume that the box is a cube). The box is open (no top) thus, total surface area of the box (4 sides and the base) is given by A = x2 + 4xy. From the problem statement, we know that A = 1200 cm2 thus, we have the following constraint relationship 1200 = x2 + 4xy. (1) The volume of the box is given by V = x2 y. Solving (1) for y and substituting it into (2) yields 1200 − x2 2 . V =x 4x (2) (3) To find the maximum volume, we take a derivative of (3) with respect to x and set the result equal to zero, to obtain 3x2 300 − = 0. (4) 4 Solving (4) for x, yields x = 20 cm. To find the y value, we substitute x = 20 cm into (1), to obtain y = 10 cm. Finally, substituting x = 20 cm and y = 10 cm into (2) yields V = 4000 cm3 .